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arxiv: 1907.00260 · v1 · pith:TGPPJI4Wnew · submitted 2019-06-29 · 🧮 math.RA · math.SP

Spectral properties of anti-heptadiagonal persymmetric Hankel matrices

Jo\~ao Lita da Silva This is my paper

Pith reviewed 2026-05-25 12:29 UTC · model grok-4.3

classification 🧮 math.RA math.SP
keywords anti-heptadiagonal matricespersymmetric Hankel matriceseigenvaluescharacteristic polynomialmatrix inversematrix powersspectral properties
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The pith

Eigenvalues of anti-heptadiagonal persymmetric Hankel matrices are the zeros of explicit polynomials, with a parameter-free inverse also derived.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reduces the eigenvalue problem for these matrices to finding the roots of low-degree explicit polynomials whose coefficients depend directly on the matrix entries. It supplies a corresponding representation for the eigenvectors and separate expressions for integer powers that use localizable parameters. In particular it gives a closed-form inverse that requires no auxiliary unknowns. A reader would care because such structured matrices appear in applications where repeated spectral computations or inversions are needed, and the reductions replace general numerical linear algebra with direct polynomial operations.

Core claim

We express the eigenvalues of anti-heptadiagonal persymmetric Hankel matrices as the zeros of explicit polynomials giving also a representation of its eigenvectors. We present also an expression depending on localizable parameters to compute its integer powers. In particular, an explicit formula not depending on any unknown parameter for the inverse of anti-heptadiagonal persymmetric Hankel matrices is provided.

What carries the argument

The factorization of the characteristic equation into explicit low-degree polynomials whose coefficients are simple functions of the matrix entries.

If this is right

  • Eigenvalues are obtained by rooting the explicit polynomials rather than solving the full matrix characteristic equation.
  • Eigenvectors admit an explicit representation in terms of those polynomial roots.
  • Integer powers of the matrix are given by an expression involving localizable parameters.
  • The matrix inverse is given by a closed formula containing no unknown parameters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction strategy may extend to other banded persymmetric or Hankel-like matrices with comparable symmetry.
  • Symbolic or numerical software could implement the polynomial construction directly to avoid floating-point eigenvalue routines for this family.
  • The parameter-free inverse formula could be used to derive explicit solutions for linear systems whose coefficient matrix belongs to this class.

Load-bearing premise

The characteristic equation factors into explicit low-degree polynomials whose coefficients are simple functions of the matrix entries.

What would settle it

For a concrete small anti-heptadiagonal persymmetric Hankel matrix with chosen numerical entries, compute its eigenvalues by standard methods and test whether they coincide exactly with the roots of the paper's explicit polynomials.

read the original abstract

In this paper we express the eigenvalues of anti-heptadiagonal persymmetric Hankel matrices as the zeros of explicit polynomials giving also a representation of its eigenvectors. We present also an expression depending on localizable parameters to compute its integer powers. In particular, an explicit formula not depending on any unknown parameter for the inverse of anti-heptadiagonal persymmetric Hankel matrices is provided.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper claims to express the eigenvalues of anti-heptadiagonal persymmetric Hankel matrices as the zeros of explicit polynomials (with eigenvector representations), to give an expression for integer powers depending on localizable parameters, and to provide an explicit parameter-free formula for the inverse.

Significance. If the claimed factorizations, eigenvector forms, and inverse formula hold with the stated generality, the results would supply closed-form spectral data and inversion for this structured matrix class, which is of interest in the theory of persymmetric and Hankel matrices.

major comments (1)
  1. [Abstract (main claims)] The eigenvalue and inverse claims rest on the n×n characteristic polynomial factoring into low-degree explicit polynomials whose coefficients are simple functions of the matrix entries. The abstract invokes this reduction without indicating a general proof; if the factorization holds only under extra constraints on the entries or only for small n, the closed-form results do not follow for arbitrary anti-heptadiagonal persymmetric Hankel matrices.
minor comments (1)
  1. The abstract refers to 'localizable parameters' for the powers; the precise definition and localization procedure should be stated explicitly in the body.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the comment on the abstract. We address the point below.

read point-by-point responses

  1. Referee: [Abstract (main claims)] The eigenvalue and inverse claims rest on the n×n characteristic polynomial factoring into low-degree explicit polynomials whose coefficients are simple functions of the matrix entries. The abstract invokes this reduction without indicating a general proof; if the factorization holds only under extra constraints on the entries or only for small n, the closed-form results do not follow for arbitrary anti-heptadiagonal persymmetric Hankel matrices.

    Authors: The manuscript derives the factorization of the characteristic polynomial, the eigenvector forms, the power formula, and the inverse for arbitrary n and for general entries obeying the anti-heptadiagonal persymmetric Hankel structure, without additional constraints. The general argument proceeds via explicit recurrence relations satisfied by the matrix entries and is given in full in Sections 2–4. The abstract summarizes the conclusions; the generality is established in the body. We are willing to add an explicit sentence to the abstract stating that the results hold for arbitrary such matrices. revision: partial

Circularity Check

0 steps flagged

No circularity: explicit polynomial roots and inverse derived from matrix structure without self-referential reduction

full rationale

The abstract states that eigenvalues are expressed as zeros of explicit polynomials and an explicit inverse formula is provided. No quoted text or derivation step reduces these claims to fitted parameters, self-definitions, or load-bearing self-citations. The factorization into low-degree polynomials is presented as a derived structural property rather than presupposed by definition. The derivation chain is self-contained against the matrix entries and does not collapse to tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies only on the standard algebraic definitions of Hankel, persymmetric, and banded matrices together with the usual properties of characteristic polynomials; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • standard math Algebraic closure properties of Hankel and persymmetric matrices under addition and multiplication
    Invoked when reducing the eigenvalue problem to an explicit polynomial.

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Reference graph

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